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New Quantum Codes from Skew Constacyclic Codes Over a Class Of Non-Chain Rings Re, q

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Abstract

In this paper, for q = pm (p is prime) such that q ≡ 1 (mod e), we study skew constacyclic codes over a class of non-chain rings \(R_{e,q}=\mathbb {F}_{q}[u]/\langle u^{e}-1\rangle \) where m, e ≥ 2 are integers. We decompose the ring into a direct sum of local rings, and consequently, skew constacyclic codes over that ring into a direct sum of skew constacyclic codes over local rings. This decomposition yields the structure of Euclidean duals of skew constacyclic codes and further a necessary and sufficient condition to contain their duals. From an application point of view, we apply the CSS (Calderbank-Shor-Steane) construction on Gray images of dual containing skew constacyclic codes and obtain many quantum codes improving the best-known codes in the literature.

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Acknowledgements

First author is thankful to the Department of Science and Technology (DST) (under CRG/2020/005927, vide Diary No. SERB/F/6780/ 2020-2021 dated 31 December, 2020) and second author is thankful to the University Grants Commission (UGC) (under Sr. No. 2121540952, Ref. No. 20/12/2015(ii)EU-V dated 31/08/2016) for financial support. Authors would also like to thank the anonymous referee(s) and the Editor for their valuable comments to improve the presentation of the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Patna, Patna, 801 106, India

    Om Prakash, Habibul Islam & Shikha Patel

  2. I2M, CNRS, Aix-Marseille University, Centrale Marseille, Marseille, France

    Patrick Solé

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  1. Om Prakash
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Prakash, O., Islam, H., Patel, S. et al. New Quantum Codes from Skew Constacyclic Codes Over a Class Of Non-Chain Rings Re, q. Int J Theor Phys 60, 3334–3352 (2021). https://doi.org/10.1007/s10773-021-04910-0

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